moving force identification: optimal state estimation approach

Journal of Sound and <ibration (2001) 239(2), 233}254doi:10.1006/jsvi.2000.3118, available online at http://www.idealibrary.com on

MOVING FORCE IDENTIFICATION: OPTIMAL STATEESTIMATION APPROACH

S. S. LAW

Civil and Structural Engineering Department, ¹he Hong Kong Polytechnic ;niversity, Hunghom,Kowloon, Hong Kong, People1s Republic of China. E-mail: [email protected]

AND

Y. L. FANG

Department of Flight <ehicle Design and Applied Mechanics, Beijing ;niversity of Aeronautics andAstronautics, Beijing, People1s Republic of China

(Received 10 November 1999, and in ,nal form 15 May 2000)

Existing techniques to identify moving forces exhibit the common weakness of havinglarge #uctuations in the identi"ed results. A new method of moving force identi"cation isdeveloped in this paper making use of the dynamic programming technique to overcome thisweakness. The forces in the state-space formulation of the dynamic system are identi"ed inthe time domain using a recursive formula based on several distributed sensormeasurements, and responses of the structure are reconstructed using the identi"ed forcesfor comparison. Like all inverse problems, the computation is ill-conditioned. However, thedynamic programming technique inherently provides bounds to the ill-conditioned forces,and results from the simulation studies and laboratory work show great improvements overexisting methods in the accuracy of identi"cation.

( 2001 Academic Press

1. INTRODUCTION

The vehicle/bridge interaction forces are important for bridge design as they contribute tothe live load component in the bridge design code. Direct measurement of the forces usinginstrumented vehicles is expensive and is subjected to bias [1, 2], while results fromcomputation simulations are subjected to modelling errors [3}5]. Inclusion of thein#uencing parameters in the model would make it computationally expensive. Systemshave been developed for weigh-in-motion of the vehicles [6, 7], but they all measure onlythe static axle loads. A technique to determine the vehicular loads from the vibrationresponses of the bridge deck is required such that the di!erent parameters of the bridge andvehicle system are accounted for in the measured responses, and the cost involved would bemuch less than that by direct measurement.

Some researchers identi"ed forces acting at a "xed location. Whiston [8] and Jordan andWhiston [9] used the arrival-time di!erence between the maximum and minimumfrequency components in the #exural wave in a beam to calculate a preliminary estimationof the force, and a further re"nement process is conducted to reconstruct the impact historyiteratively so that the force would not have a negative value. Michaels and Pao [10]developed a method that determined an oblique dynamic force using wave motiondisplacement measurements.

0022-460X/01/020233#22 $35.00/0 ( 2001 Academic Press

234 S. S. LAW AND Y. L. FANG

Other researchers worked on the problem in which both the force history and its locationand unknown. Examples include using the modal response data to determine the location ofimpact forces on the read/write head of computer disks [11]. Doyle [12] also developeda method for determining the location and magnitude of an impact force using the phasedi!erence of the signals measured at two di!erent locations straddling the impact point.Flexural wave propagation response was used to determine the location of the structuralimpacts.

All the above works are based on the propagating-wave approach which relies on modelsin the frequency domain. Busby and Trujillo [13] reconstructed the force history usinga standing wave approach and Hollandsworth and Busby [14] veri"ed this experimentallywith a force applied at a known location and accelerometers were used as sensors. Simonian[15, 16] used a dynamic programming "lter to predict wind loads on a structure. Druz et al.[17] formulated a non-linear inverse problem and tried to "nd the location and magnitudeof the external force. This force, however, was not general and is con"ned to a sinusoidalfunction de"ned by its amplitude and phase.

Research on the identi"cation of moving forces just started a few years ago. The timedomain approach [18] models the structure and forces with a set of second orderdi!erential equations. The forces are modelled as step functions in a small time interval.These equations of motion are then expressed in the modal co-ordinates, and they aresolved by convolution in time domain. The forces are then identi"ed using the modalsuperposition principle. The frequency and time domain approach [19] performs Fouriertransformation on the equations of motion, which are expressed in modal co-ordinates. TheFourier transforms of the responses are expressed in terms of those for the forces, and thetime histories of the forces are found directly by the least-squares method. The modalapproach [20] identi"es the forces completely in the modal co-ordinates. Measureddisplacements are converted into modal displacements with an assumed shape function.The modal velocities and accelerations are then obtained by di!erentiation. The forces arethen identi"ed by solving the uncoupled equations of motion in modal co-ordinates.

All the three approaches require the computation of matrix inverse, and they arecomputationally expensive. These approaches can be shown to be numericallyill-conditioned at the start and end of the time histories.

This paper further explores the area of moving force identi"cation making use of thedynamic programming technique. The forces are identi"ed in the time domain usingrecursive formula, and responses of the structure are reconstructed using the identi"edforces for comparison. The forces are identi"ed based on several distributed sensormeasurements. Like all inverse problems, the computation is ill-conditioned. However,the dynamic programming technique inherently provides bounds to the ill-conditionedforces, and results from both the simulation studies and laboratory work show greatimprovements in the accuracy of identi"cation over existing methods. The simulationstudies are on the identi"cation of a single force and two forces moving on a simplysupported beam. The experiment study has a model car moving on a simply supportedbeam, and the interaction forces are identi"ed from velocity and bending moment responsesof the beam.

2. ASSUMPTIONS

The following assumptions are made on the dynamic system model:

1. The changes in the system characteristics, i.e., the sti!ness, damping and mass matricesunder the passage of the force are negligible.

MOVING FORCE IDENTIFICATION 235

2. Structural damping is included in the analysis.3. The structure may not be at rest before the application of the load.4. There is no restriction on the type of force history to be identi"ed.5. The Euler}Bernoulli beam model is used with the shear e!ect neglected.

3. NODAL FORCES FROM AN APPLIED FORCE

When a force time history f1

is applied on a two-dimensional "nite beam element oflength l between the ith and (i#1)th nodes at a point distance x from the left end, the nodalforces at each end of the beam element can be represented by

Ri"A1!

3x2

l2#2

x3

l3 B f1,

Mi"Ax!

2x2

l#

x3

l2B f1,

Ri`1

"A3x2

l2!

2x3

l3 B f1,

Mi`1

"Ax3

l2!

x2

l B f1, (1)

where Ri, R

i`1are the vertical nodal forces, and M

i, M

i`1are the nodal bending moments

at the ith and (i#1)th node of the structure respectively. These nodal forces are groupedinto the global force vector as

P"Y(x) ) f1, (2)

where P is the nodal force vector and Y(x) is the vector on the location of the applied force.For the case of multi-forces acting on the beam element, the global force vector arising fromthe ith force is represented by

Pi"Y(x

i) ) f

i. (3)

4. STATE-SPACE MODEL FORMULATION

The "nite element representation of an n-d.o.f.s dynamic system is given by

MuK#Cu#Ku"P, (4)

where u is a vector containing all the displacements of the model, uR is the "rst derivative ofu with respect to time t, M is the system mass matrix, C is the system damping matrix, K isthe system sti!ness matrix, and P represents the system of exciting forces which is a functionof the location and magnitude of the applied forces as shown in equation (3).

Using the state-space formulation, equation (4) is converted into a set of "rst orderdi!erential equations as follows:

X0 "K*X#P1 , (5)


where

X"Cu

u5 D2n]1

, K*"C0

!M~1K

I

!M~1CD2n]2n,

P1 "C0

!M~1PD2n]1

"C0

!M~1YDn]nf

fnf]1 , (6)

where X represents a vector of state variables of length 2n containing the displacements andvelocities of the nodes, n

fis the number of forces, and f is a vector of length n

frepresenting

the unknown applied forces. These di!erential equations are then rewritten as discreteequations using the standard exponential matrix representation.

Xj`1

"FXj#G1

j`1P1j, (7)

F"eK*h (8)

and

G1 "K*~1(F!I), (9)

where matrix F is the exponential matrix, and together with matrix G1 it represents thedynamics of the system, ( j#1) denotes the value at the ( j#1)th time step of computation,the time step h represents the time di!erence between the variable states X

jand X

j`1in the

computation, and G1 is a matrix relating the forces to the system. Substituting equations (6)and (9) into equation (7) we have

Xj`1

"FXj#G

j`1fj, (10)

where

G"G1 2n]2n C0

!M~lYD2n]nf

. (11)

5. PROBLEM STATEMENT

The problem we have is one in which the system matrices K, C and M are known togetherwith information on some of the displacements and velocities. However, the forcing term f isunknown. The goal of this problem is to "nd the forcing term f that causes the systemdescribed in equation (10) to best match the measurement.

In practice, it is not possible to measure all the displacements and velocities, and, onlycertain combinations of the variables X

jare measured. The measurement equation is given

as

dj"QX

j, (12)

where djis an (m]1) measurement vector, Q is an (m]2n) selection matrix relating the

measurements to the state variables, and Xj

is of dimension (2n]1). The actual


measurements are represented by a vector Zjwhich is of the same dimension as d

j. The

number of measured variables m is usually much less than the number of state variables (orn d.o.f.s of the system) but greater than or equal to n

f, the length of vector f. In the case of

a two-dimensional simply supported beam divided into ¸ elements, n"2(¸#1)!2including all vertical displacements and rotational displacements at each of its nodes.

When the unknown force fjis included in equation (10), an exact match of the model with

the measured data is usually not possible. This is due to the fact that all measurements havesome degree of noise. Even the least-squares criterion is not su$cient becausea mathematical solution that will minimize the least-squares error E represented by

E"

N+j/I

((Zj!QX

j), A(Z

j!QX

j)) (13)

will usually end up with the model exactly matching the data. (x, y) in equation (13) denotesthe inner product of two vectors x and y. This situation could be avoided by addinga smoothing term to the least-squares error [21, 22] to become a non-linear least-squaresproblem

E"

N+j/1

((Zj!QX

j), A (Z

j!QX

j)#( f

j, Bf

j)). (14)

The second term is known as the regularization parameter and the method is called theTikhonov [23] method. Matrices A(m]m) and B (n

f]n

f) are symmetric positive-de"nite

weighting matrices that provide the #exibility of weighting the measurements and theforcing terms. Matrix A is usually an identity matrix and matrix B is a diagonal matrix. Thesecond term with the positive parameter B has the e!ect of smoothing the identi"ed forces.A small value of B causes the solution to match the data closely but produces largeoscillatory deviations. A large value of B produces smooth forces that may not match thedata well. When B is zero, the solution becomes that for the least-squares problem.

6. STRAIN MEASUREMENTS

For a two-dimensional "nite beam element of length l represented by the degrees offreedom (u

1, h

1, u

2, h

2) at its two ends, the strain at any cross-section distance x from the left

end of the beam element can be given in terms of the d.o.f.s. at its two ends as

ex"A

!y

l 3 B (12x!6l)u1!l (6x!4l )h

1!(12x!6l )u

2#l (6x!2l )h

2], (15)

where y represents the distance from the neutral axis of the beam. In the veri"cation of thisstudy, the strain measurements and hence the bending moments are related to all of thetransverse displacements and rotational displacements at the ends of the element using thisrelationship.

7. MATRIX G FOR TWO MOVING FORCES WITH KNOWN SPEED

Suppose that we have two forces spaced at a constant distance moving across a simplysupported beam at a constant speed c. The matrices G1 and G in equations (9) and (11),


respectively, vary for di!erent locations of the forces. Rewrite equations (2) and (9) as

P"[Y(x1), Y(x

2)] A

f1

f2B , (16)

G1 "CG1

11G1

21

G112

G122D (17)

and equation (6) becomes

P1 "C0

!M~1Y(x1)

0

!M~1Y(x2)DA

f1

f2B . (18)

The discrete representation of the system in equation (7) can then be written as

Xj`1

"FXj#G1

j`1P1j

"FXj#C

G111

G121

G112

G122D C

0

!M~1Y(x1)

0

!M~1Y(x2)D A

f1f2B

(19)

"FXj#C

!G112

M~1Y(x1)

!G122

M~1Y(x1)

!G112

M~1Y(x2)

!G122

M~1Y(x2)DA

f1f2B

"FXj#G

j`1(x

1, x

2) A

f1f2B

and in matrix form as

Xj`1

"FXj#G

j`1fj, (20)

where Gj`1

is the value at the ( j#1)th time step. Note the equation (20) is the same asequation (10) but for two moving forces.

8. DYNAMIC PROGRAMMING

To minimize the least-squares error E in equation (14) over the sequence of the forcingvector f

j, the dynamic programming method [24] and Bellman's Principle of Optimality

[25] are applied. The minimum value of E at the nth stage for any initial state X is written as

gn(X)"min

fj

En(X, f

j) . (21)

A recursive formula for equation (21) is derived from Bellman's Principle of Optimality as

gn~1

(X)"minfn~1

((Zn~1

!QX), A (Zn~1

!QX)#(fn~1

, Bfn~1

)#gn(FX#G

nfn~1

)). (22)

This equation represents the classic dynamic programming structure in that the minimumat any point is determined by selecting the decision f

n~1to minimize the immediate cost (the

"rst and second terms) and the remaining cost resulting from the decision (the third term). Itis noted that the minimization is performed over a previously determined function g

n. The


terms fn

and gn

are the optimal forcing term and the optimal cost term respectively. Thesolution is obtained by starting at the end of the process, n"N, and working backward ton"1. At the endpoint, the minimum is determined from

gN(X)"min

fN

[(ZN!QX), A(Z

N!QX)#f

N, Bf

N]. (23)

With fN"0, we obtain the minimum solution as shown in equation (24) by expanding

equation (23)

gN(X)"q

N#(X, S

N)#(X, R

NX), (24)

where

qN"Z

N, AZ

N,

SN"!2QT

NAZ

N,

RN"QT

NAQ

N. (25)

These are the initial conditions for working backward at n"N. Substituting equation (24)for the nth and (n!1)th steps into equation (22), and expanding the right-hand side of theequation, we have

qn~1

#(X, Sn~1

)#(X, Rn~1

X)"minfn~1

[( fn~1

#<nX#U

n), H

n(fn~1

#VnX#U

n)#r

n~1(X)],

(26)

where

Hn"B#GT

nR

nG

n, 2H

nVn"2GT

nR

nF, V

n"H~1

nGT

nR

nF,

2HnU

n"GT

nSn, U

n"(H~1

nGT

nSn)/2.

rn~1

(X)"C(q

n#ZT

n~1AZ

n~1)#XT(QTAQ#FTR

nF )X#XT (FTS

n!2QTAZ

n~1)

!XTVTnH

nV

nX!UT

nH

nU

n!2XTVT

nH

nU

nD .

Minimizing the term on the right-hand side of equation (26) yields the optimal forcing termas

fn~1

"!H~1n

GTn CRn

FXn~1

#

Sn

2 D (27)

and equation (26) becomes

qn~1

#(X, Sn~1

)#(X, Rn~1

X)"rn~1

(X). (28)

Equating like powers of X in equation (28) will yield the following relationship:

Rn~1

"QTAQ#FT[I!RTnG

nH~1

nGT

n]R

nF,

(29)Sn~1

"!2QTAZn~1

#FT[I!RTnG

nH~1

nGT

n]S

n.


These are the recursive formulae required to determine the optimal solution of equation(22). The complete sequence of operations is as follows:

Step 1: Matrices Q and Z and the speed of the forces are obtained from measurement;Step 2: Matrix G1 and hence matrix G are obtained from information on the location of the

forces from equations (9) and (19);Step 3: Compute the initial values q

N, R

Nand S

Nfrom equation (25); compute H

Nfrom

equation (26);Step 4: Compute S

n~1and R

n~1from equation (29) for n"N to 1;

Step 5: Initial condition of X is set as zero and compute the responses Xj`1

from equation(20) for j"0 to N, and compute the forces f

n~1from equation (27) for n"1 to N;

Step 6: Steps 1}5 are repeated for a di!erent smoothing parameter B. Convergence isreached when the error computed from equation (30) is reduced to a predeterminedvalue.

9. SIMULATION AND RESULTS

The proposed method is studied for its accuracy and e!ectiveness in identifying thefollowing simulated forces

(a) for single moving force identi"cation

f1(t)"40 000 [1#0)1 sin (10nt)#0)05 sin (40nt)]N;

(b) for two moving forces identi"cation

f1(t)"20 000 [1#0)1 sin (10nt)#0)05 sin (40nt)]N,

f2(t)"20 000 [1!0)1 sin (10nt)#0)05 sin (50nt)]N.

The two forces are at a constant spacing of 4 m apart and they are moving together ona simply supported beam at a velocity of 40 m/s. The physical parameters of the beam are:

EI"1)274916 * 1011Nm2, oA"12 000 kg/m, ¸"40 m.

The lowest three natural frequencies of the beam are 3)2, 12)8 and 28)8 Hz. The "niteelement model of the beam consists of 10 elements with 11 nodes and 20 rotational andtranslational d.o.f.s. The state variable matrix X has a dimension of (40]1). A samplingfrequency of 200 Hz is used indicating that 100 Hz is the upper frequency limit of the study.The time when there are forces on the beam is 1)1 s, and 220 data points are used.

Dynamic analysis was performed on this system to "nd the velocity and bending momenttime histories at speci"ed locations. Five per cent root-mean-square normally distributedrandom noise with zero mean and unit standard deviation was added to these responses tosimulate the polluted measurements.

The error in the force identi"cation is calculated by the following equation where EfE isthe norm of a matrix:

Error"Ef

identified!f

trueE

Eftrue

E]100%. (30)

Since the true force is known, the optimal regularization parameter B is obtained bycomparing the true force f

truewith the identi"ed values f

identified, and an error curve can be

TABLE 1

Error in one force identi,cation (in per cent) (with 5% noise in response)

Sensor location andresponse type Moment Velocity Both

1/2 span 26)8 24)5 26)91/4 span 26)1 21)7 26)01/2 span, 1/4 span 26)7 19)8 26)6


plotted for di!erent values of B. The error is calculated from equation (30). It is noted thatthe optimal value of B corresponds to the smallest error.

If both the measured bending moments and velocities are used together to identify themoving forces, the velocity component and the bending moment component in the vectorX in equation (10) should be scaled by their respective norms to have dimensionless units.

9.1. SINGLE-FORCE IDENTIFICATION

Nine combinations of the responses at 1/4 and 1/2 span as shown in Table 1 are used inthe identi"cation. The errors in Table 1 show that the use of single or multiple responsesdoes not cause signi"cant di!erence in the errors of identi"cation. Results not shown hereindicate that the use of an additional sensor at 3/4 span does not improve signi"cantly theidenti"ed results.

The time histories and the PSDs of the calculated and true forces for cases of (1/4v, 1/2v)and (1/4m, 1/4v) are shown in Figure 1. The time histories closely match each other exceptat the start and end of the time duration. It is found that the variation in the identi"ed forcesincreases with decrease in B. An inspection of the PSDs of the force reveals that goodmatching between the forces is found around the exciting frequencies but with large noise inthe upper frequency range. The random noise in the polluted responses is re#ected in theupper frequency range of the identi"ed forces with little adverse e!ect on the results in thelower frequency range. The use of bending moments seems to cause poorer performancescompared with the velocities.

The reconstructed bending moment and velocity at 1/4 span are compared with themeasured ones in Figures 2 and 3. The time histories match closely with the true ones. ThePSDs of the responses indicate that errors exist in the upper frequency range which is againsuspected to be due to the simulated noise e!ect.

9.2. TWO-FORCES IDENTIFICATION

Bending moment and/or velocity responses at 1/4, 1/2 and 3/4 spans in 20 combinationsas shown in Table 2 are used to identify the two forces. The results are obtained in a mannersimilar to the single-force identi"cation.

The errors between the calculated and the true forces are shown in Table 2 for di!erentsensor combinations. The errors calculated by the time domain method (TDM) [18] arealso included in brackets for comparison. Note that the TDM uses accelerationmeasurements instead of velocity measurements in the identi"cation. The proposed methodin general gives far smaller error than the time domain method. It is only in the case of using

Figure 1. Identi"ed single force and power spectrum from simulation: **, true; ) ) ) ) ) , 1/4v, 1/2v; } } }, 1/4m,1/4v.

Figure 2. Comparison of velocity at 1/4 span: **, true; ) ) ) ) ) , 1/4v, 1/2v; } }}, 1/4m, 1/4v.


Figure 3. Comparison of bending moment at 1/4 span: **, true; ) ) ) ) ) , 1/4v, 1/2v; } } }, 1/4m, 1/4v.

TABLE 2

Error in two-forces identi,cation (in per cent) (with 5% noise in response)

Sensor location and response type First force Second force Total force

1/4m, 1/2m 35)6s 37)4s 24)31/4v, 1/2v 47)5(222) 48)7(78)6) 33)51/2m, 1/2v 36)3s 38)4s 24)51/4m 1/4v 38)7s 40)6s 27)31/2m, 1/4v 36)4s 38)4s 24)11/4m, 1/2m, 3/4m 32)1s 33)1s 20)61/4v, 1/2v, 3/4v 38)9(10)7) 39)8(10)7) 27)51/4m, 1/2m, 1/4v 35)6s 37)4s 24)31/2m, 3/4m, 1/4v 33)5 34)1 21)51/2v, 3/4m, 3/4v 35)5 37)7 25)81/4m, 1/2m, 1/2v 35)6s 37)4(474) 24)31/4m, 1/4v, 1/2v 38)7(206) 40)6(156) 27)31/2m, 1/4v, 1/2v 36)3(193) 38)4(46)8) 24)51/2m, 1/4v, 3/4v 36)4 38)4 24)11/4m, 1/2m, 1/4v, 1/2v 35)6(201) 37)4(49)7) 24)01/4m, 1/2m, 1/2v, 3/4v 32)1 33)2 20)71/2m, 1/4v, 1/2v, 3/4v 36)4 38)4 24)11/4m, 1/2m, 3/4m, 1/4v, 1/2v 32)1 33)2 20)71/4m, 1/2m, 1/4v, 1/2v, 3/4v 35)6 37)3 24)01/4m, 1/2m, 3/4m, 1/4v, 1/2v, 3/4v 32)1 33)1 20)6

sIndicates error larger than 1000%.


Figure 4. Identi"ed "rst force and power spectrum from simulation:**, true; ) ) ) ) ) , 1/4v, 1/2v, 3/4v; } } }, 1/4m,1/2m, 3/4m.


three acceleration measurements that the TDM gives better results than the present caseusing three velocities. The results shown in Table 2 indicate that (1) many sensorcombinations would give similar errors in the identi"ed forces; (2) the use of more than twosensors may not improve the result; and (3) both velocity and bending moment giveapproximately the same accuracy in the identi"ed forces.

The time histories and PSDs of the identi"ed forces from using (1/4v, 1/2v, 3/4v) and(1/4m, 1/2m, 3/4m) sensor combinations are shown in Figures 4}6. All the identi"ed forcesvary closely with the true force in the middle length of the time duration between 0)25 and0)75 s. The results have a lot of variations due to the simulated noise e!ect. Other results notshown here using di!erent sensor combinations also exhibit a similar pattern in theidenti"ed forces. Note that the discrepancies at the start and end of the time historiescontribute greatly to the overall error.

The above results show that accuracy in two-forces identi"cation is lower than that insingle-force identi"cation. This is due to a component in the simulated individual forceswith same amplitude and opposite phase. This results in large errors in the time histories.The combined force shown in Figure 6 indicates that most of these errors are cancelled outleading to a greatly improved time history and reduced error in the combined identi"edforce as shown in Table 2.

10. EXPERIMENT AND RESULTS

The experimental set-up is shown diagrammatically in Figure 7. The main beam3376 mm long with 100 mm]25 mm uniform cross-section is simply supported.

Figure 5. Identi"ed second force and power spectrum from simulation:**, true; ) ) ) ) ) , 1/4v, 1/2v, 3/4v; } } },1/4m, 1/2m, 3/4m.

Figure 6. Identi"ed combined force and power spectrum from simulation:**, true; ) ) ) ) ) , 1/4v, 1/2v, 3/4v; } } },1/4m, 1/2m, 3/4m.


Figure 7. Diagramatic drawing of experimental set-up.


A U-shaped aluminium section on the upper surface of the beams served as direction guidefor the car. An electric motor is used to pull a model car along the guide with a string woundon the shaft of the motor. The rotating speed of the motor can be adjusted for di!erentspeeds. Seven photoelectric sensors are mounted on the beams to monitor the moving speedof the car. The second sensor is located at the point where the front wheels of the car just geton the main beam, and the last sensor is located at the point where the rear set of wheels justget o! the main beam. They are used to measure the speed of the model car.. The others arelocated on the main beam at a spacing of 0)776m to check on the uniformity of the speed.

Three strain guages and four accelerators are mounted at the bottom of the main beam tomeasure the responses. One gauge and one accelerator are mounted at each of thecross-sections at the 1/4, 1/2 and 3/4 spans. The fourth accelerator is mounted at the 3/8span. An eight-channel dynamic testing and analysis system (DTAS) is used for datacollection in the experiment. The "rst channel is used to monitor the signal of thephotoelectric sensors. The second, third and fourth channels are used to measure the signalof the strain gauges. The remaining channels are used to measure the acceleration responses.The sampling frequency is 128 Hz.

The model car has two axles at a spacing of 0)203 m and in runs on four rubber wheels.The mass of the whole car is 7)1 kg. The bouncing, pitching and rolling natural frequenciesof the car are 27)5, 42)9 and 69)4 Hz respectively. The "rst three natural frequencies of thebeam are 6)6, 18)5 and 39)5 Hz. The average speed of the car is 3)102 m/s in the presentstudy.

In this practical case when ftrue

is not known, the seminorm of the solution against theregularization parameter B is plotted. The seminorm of the estimated forces is

E1"Ef (identify)j`1

!f (identify)j

E, (31)

where f (identify)j

, f (identify)j`1

are the identi"ed forces with Bjand B

j#DB. The value of B which

corresponds to the smallest seminorm is the optimal value.


10.1. EXPERIMENTAL PROCEDURES

Before the actual test on the experimental set-up, signals from the strain gauges wereinspected and the zero-shift components were identi"ed. The signals were then calibrated byadding masses at the middle of the main beam. During the calibration, the signals of thestrain gauges were found to be not very stable and repeatability was not perfectlysatisfactory. It is noted that this will lead to calibration errors in the identi"ed results. Thecar was placed at the right end of the leading beam, and the DTAS was set in pre-triggerstate at channel 1. Power for the motor was turned on, and the car moved on top of thebeams. Eight channels of signal were acquired. The uniformity of the speed was checked. Ifthe speed was stable, the above steps were repeated to check whether the properties of thestructure and measurement system were changed or not. If no signi"cant change was found,the recorded data were accepted. The zero-shifts in the measured signals were removed, andthe signals were calibrated with measured channel sensitivities. The time histories betweentwo measured points of the responses were sub-divided into 10 sub-divisions such that thetime di!erence between two time steps is 7)8125 * 10~4 s. A smaller sampling time interval isimportant for the accuracy of the iterative computation. The measured acceleration recordswere integrated to velocity records with an algorithm developed by Petrovski andNaumovski [26] with a low-pass "lter at 0)0625 Hz.

10.2. SINGLE-FORCE IDENTIFICATION

Nine combinations of the measured responses at 1/4 and 1/2 spans are used to identifythe forces as a single force. Correlation coe$cients between the reconstructed responses

Figure 8. Identi"ed single force and power spectrum from experiment:**, static force; } } } , 1/4m, 1/2m, ) ) ) ) ),1/4v, 1/2v.

Figure 9. Comparison of velocity at 3/4 span in experiment:**, measured; } } } , 1/4 m, 1/2m; ) ) ) ) ), 1/4v, 1/2v.

Figure 10. Comparison of bending moment at 3/4 span in experiment:**, measured; } } } , 1/4 m, 1/2m; ) ) ) ) ),1/4v, 1/2v.


TABLE 3

Correlation coe.cients between measured and reconstructed responses 2 single-forceidenti,cation

Combinations of the Comparing moment at Comparing velocityresponses 3/4 span at 3/4 span

1/2m 0)349 0)1551/4m 0)801 0)6001/4v 0)369 0)8691/2v 0)200 0)2541/2m, 1/4m 0)939 0)7441/2v, 1/4v 0)364 0)9461/4m, 1/2v 0)814 0)6291/4m, 1/4v 0)796 0)5941/2m, 1/4v 0)520 0)142

Figure 11. Identi"ed forces from sets of two responses from experiment: **, static force; } }} , 1/4m, 1/2m,) ) ) ) ), 1/4v, 1/2v.


Figure 12. Identi"ed forces from sets of three responses from experiment:**, static force; } } } , 1/4m, 1/2m,3/4m; ) ) ) ) ), 1/4v, 1/2v, 3/4v.


using the identi"ed forces and measured responses at 3/4 span are calculated to evaluate theaccuracy of the identi"ed force. Clearly, a larger coe$cient means that the identi"ed forcesare more accurate than that with a smaller coe$cient, but the larger coe$cientdoes not mean the identi"ed force is accurate enough because this indirect method ofchecking on the identi"ed results is not fully su$cient. The coe$cients are shownin Table 3, and some of the identi"ed results are shown in Figures 8}10. It is notpossible to identify the static component of the forces using only velocity measurements,and the static forces are added to the identi"ed forces in the "gures for convenience ofcomparison.

The results in Table 3 show that the use of bending moments both at 1/2 span and at 1/4span is most suitable for single moving force identi"cation. Figures 9 and 10 show that thePSDs of the responses match closely, although the time histories are not very close to eachother. It is suspected that the calibration errors lead to these di!erences in the time histories.

Figure 13. Comparison of velocity at 3/8 span using two responses in experiment:**, measured; } } } , 1/4m,1/2m, ) ) ) ) ), 1/4v, 1/2v.


10.3. TWO-FORCES IDENTIFICATION

Twelve combinations of the measured responses are used to identify the two forcesseparately. The velocity of the main beam at the 3/8 span is reconstructed using theidenti"ed forces, and it is compared with the measured response.

Some of the identi"ed results are shown in Figures 11}14. The component with the sameamplitude and opposite phase in the two identi"ed forces in Figures 11 and 12 is due to thepitching motion of the model car. The two identi"ed forces are added to obtain a resultantforce as shown in the "gures. This resulting force can be used as a good estimate of the totalequivalent static load of the vehicle.

Figures 13 and 14 show that those curves obtained from using velocity responses closelymatch the measured responses, while those obtained from using bending moment responseshave a large di!erence in the amplitude. This again leads to the suspicion of the existence ofcalibration error in the strain measurements.

Correlation coe$cients are calculated between the reconstructed and measuredresponses and they are shown in Table 4. They show that (1) velocity response alone givesbetter results for the two-forces identi"cation; and (2) 1/4 span responses are better than theother responses. Comparison of the correlation coe$cients in Tables 3 and 4 andcomparing the responses in Figures 9}12 show that results from the identi"cation as twomoving forces are more accurate than those from the identi"cation as a single force.

11. DISCUSSIONS

The ill-conditioned identi"ed forces have large #uctuations in their time histories. Theirmagnitude are, however, bounded with the use of non-linear least-squares minimization as

Figure 14. Comparison of velocity at 3/8 span using three responses in experiment:**, measured; } } } , 1/4m,1/2m, 3/4; ) ) ) ) ), 1/4v, 1/2v, 3/4v.

TABLE 4

Correlation coe.cients between measured and reconstructedvelocity at 3/8 span 2 two-forces identi,cation

Response combinations Correlationlocation and type coe$cient

1/2m, 1/4m 0)7421/2m, 1/4m, 3/4m 0)7481/2v, 1/4v 0)9891/2v, 1/4v, 3/4v 0)9881/2m, 1/2v 0)0181/2m, 1/4m, 1/2v 0)7431/2m, 1/4m, 1/2v, 1/4v 0)7441/4m, 1/4v 0)7391/4m, 1/4v, 1/2v 0)7471/2m, 1/4v 0)2361/2m, 1/4v, 1/4m 0)7431/4v, 1/2v, 1/2m 0)245


shown in equation (14), and the errors computed for the identi"ed forces are much smallerthan those obtained from TDM [18] without regularization. However, the short periods atthe start and end of the time histories still contribute greatly to the total error as computedfrom equation (30). This error can be reduced as reported by Choi and Chang [27] by using


di!erent smaller B matrix at these two time durations. The forces identi"ed from theexperimental data are fairly smooth varying around the static force, and the identi"edcombined force can be useful as a reliable estimate of the static weight of the vehicle crossingthe bridge.

12. CONCLUSIONS

A method based on dynamic programming is developed to identify moving forces frommeasured responses of a simply supported beam. This method provides bounds to theidenti"ed forces in solving the ill-conditioned problem, and the errors of identi"cation aremuch smaller than those obtained from the existing time domain method in comparing theidenti"ed forces from using di!erent combinations of measured responses in bothsimulation and laboratory studies. The state-space formulation of the dynamic system canbe extended to include a more complicated "nite element model of a structure undermulti-forces excitation.

ACKNOWLEDGMENTS

The work described in this paper was supported by a grant from the Hong KongPolytechnic University Research Funding Project No. S571.

REFERENCES

1. R. CANTINENI 1992 Swiss Federal ¸aboratories for Materials ¹esting and Research (EMPA)Report No. 220, 240p. Dynamic behaviour of highway bridges under the passage of heavy vehicles.

2. R. J. HEYWOOD 1994 International Journal of<ehicle Design. In#uence of truck suspensions on thedynamic response of a short span bridge.

3. M. F. GREEN and D. CEBON 1994 Journal of Sound and<ibration 170, 51}78. Dynamic response ofhighway bridges to heavy vehicle loads: theory and experimental validation.

4. Y. B. YANG and J. D. YAU 1997 Journal of Structural Engineering, ASCE 123, 1512}1518.Vehicle}bridge interaction element for dynamic analysis.

5. K. HENCHI, M. FAFARD and G. DHATT 1998 Journal of Sound and <ibration 212, 663}683. Ane$cient algorithm for dynamic analysis of bridges under moving vehicles using a coupled modaland physical components approach.

6. R. J. PETERS 1984 Proceedings of 12th ARRB Conference, Vol. 12, 10}18. A system to obtainvehicle axle weights.

7. R. J. PETERS 1986 Proceedings of 13th ARRB and 5th REAAA Combined Conference, Part 6,70}83. An unmanned and undetectable highway speed vehicle weighing system.

8. G. WHISTON 1984 Journal of Sound and <ibration 97, 35}51. Remote impact analysis by use ofpropagated acceleration signals, I: theoretical methods.

9. R. JORDAN and G. WHISTON 1984 Journal of Sound and <ibration 97, 53}63. Remote impactanalysis of use of propagated acceleration signals, II: comparison between theory and experiment.

10. J. MICHAELS and Y. H. PAO 1986 Journal of Applied Mechanics 53, 61}68. Determination ofdynamic forces from wave motion measurements.

11. J. F. DOYLE 1984 Experimental Mechanics 24, 265}270. Further developments in determining thedynamic contact law.

12. J. F. DOYLE 1987 Experimental Mechanics 27, 229}233. An experimental method for determiningthe location and time of initiation of an unknown dispersing pulse.

13. H. R. BUSBY and D. M. TRUJILLO 1987 Computers and Structures 25, 109}117. Solution of aninverse dynamics problem using an eigenvalue reduction technique.

14. P. E. HOLLANDSWORTH and H. R. BUSBY 1989 International Journal of Impact Engineering 8,315}322. Impact force identi"cation using the general inverse technique.


15. S. S. SIMONIAN 1981a International Journal of Numerical Method in Engineering 17, 357}365.Inverse problems in structural dynamics*I. Theory.

16. S. S. SIMONIAN 1981b International Journal of Numerical Method in Engineering 17, 367}386.Inverse problems in structural dynamics*II. Applications.

17. J. DRUZ, J. D. CRISP and T. RYALL 1991 AIAA Journal 29, 464}470. Determining a dynamic forceon a plate*an inverse problem.

18. S. S. LAW, T. H. T. CHAN and Q. H. ZENG 1997 Journal of Sound and<ibration 201, 1}22. Movingforce identi"cation: a time domain method.

19. S. S. LAW, T. H. T. CHAN and Q. H. ZENG 1999 Journal of Dynamic Systems, Measurement andControl ASME 12, 394}401. Moving force identi"cation: a frequency and time domains analysis.

20. T. H. T. CHAN, S. S. LAW, T. H. YUNG and X. R. YUAN 1999 Journal of Sound and <ibration 219,503}524. An interpretive method for moving force identi"cation.

21. H. R. BUSBY and D. M. TRUJILLO 1993 Inverse Problems in Engineering: ¹heory and Practice.ASME Press, USA, 155}161. An inverse problem for a plate under pulse loading.

22. J. C. SANTANTAMARINA and D. FRATTA 1998 Introduction to Discrete Signals and InverseProblems in Civil Engineering. ASCE Press, USA, 200}238.

23. A. N. TIKHONOV 1963 Soviet Mathematics 4, 1035}1038. On the solution of ill-posed problemsand the method of regularization.

24. D. M. TRUJILLO 1978 International Journal of Numerical Methods in Engineering 12, 613}624.Application of dynamic programming to the general inverse problem.

25. R. BELLMAN 1967 Introduction to the Mathematical ¹heory of Control Processes. New York:Academic Press.

26. J. PETROVSKI and N. NAUMOVSKI 1979 Report 66. I211S, Stronjze. Processing of strong motionaccelerograms. Part 1: analytical methods.

27. K. Y. CHOI and F. K. CHANG 1996 AIAA Journal 34, 136}142. Identi"cation of impact force andlocation using distributed sensors.

APPENDIX A: NOMENCLATURE

gn(X) solution corresponds to the nth stage of dynamic programming

l length of "nite beam elementu1, u

2"nite element end translations

fj

vector of force in the jth stage of forward processu vector of all displacementsu5 "rst derivative of u with respect to timeA, B weighting matricesC system damping matrixE least-squares error of the solutionF exponential matrixG system matrixK system sti!ness matrixM system mass matrixR

i, M

ithe ith nodal vertical force and moment

P global nodal force vectorQ selection matrixR

n, S

ninitial conditions in the backward process of dynamic programming

X vector of state variable of displacement and velocityY matrix of information on force locationZ measurement vector

moving force identification: optimal state estimation approach

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